## Econometric Analysis of Count DataThe “count data” ?eld has further ?ourished since the previous edition of this book was published in 2003. The development of new methods has not slowed down by any means, and the application of existing ones in applied work has expanded in many areas of social science research. This, in itself, would be reason enough for updating the material in this book, to ensure that it continues to provide a fair representation of the current state of research. In addition, however, I have seized the opportunity to undertake some major changes to the organization of the book itself. The core material on cross-section models for count data is now presented in four chapters, rather than in two as previously. The ?rst of these four chapters introduces the Poissonregressionmodel,anditsestimationbymaximumlikelihoodorpseudo maximum likelihood. The second focuses on unobserved heterogeneity, the third on endogeneity and non-random sample selection. The fourth chapter provides an extended and uni?ed discussion of zeros in count data models. This topic deserves, in my view, special emphasis, as it relates to aspects of modeling and estimation that are speci?c to counts, as opposed to general exponential regression models for non-negative dependent variables. Count distributions put positive probability mass on single o- comes, and thus o?er a richer set of interesting inferences. |

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### Contents

1 | |

2 | |

4 | |

6 | |

222 Genesis of the Poisson Distribution | 10 |

223 Poisson Process | 11 |

224 Generalizations of the Poisson Process | 14 |

225 Poisson Distribution as a Binomial Limit | 15 |

52 Incidental Censoring and Truncation | 148 |

522 Models of NonRandom Selection | 149 |

523 Bivariate Normal Error Distribution | 150 |

524 Outcome Distribution | 152 |

525 Incidental Censoring | 153 |

526 Incidental Truncation | 154 |

53 Endogeneity in Count Data Models | 156 |

532 Parameter Ancillarity | 157 |

226 Exponential Interarrival Times | 16 |

227 NonPoissonness | 17 |

23 Further Distributions for Count Data | 20 |

232 Binomial Distribution | 25 |

233 Logarithmic Distribution | 27 |

234 Summary | 28 |

24 Modiﬁed Count Data Distributions | 30 |

242 Censoring and Grouping | 31 |

243 Altered Distributions | 32 |

25 Generalizations | 33 |

252 Compound Distributions | 36 |

253 Birth Process Generalizations | 39 |

254 Katz Family of Distributions | 40 |

255 Additive LogDiﬀerenced Probability Models | 41 |

256 Linear Exponential Families | 42 |

257 Summary | 44 |

26 Distributions for Over and Underdispersion | 45 |

262 Generalized Poisson Distribution | 46 |

263 Poisson Polynomial Distribution | 47 |

264 Double Poisson Distribution | 49 |

27 Duration Analysis and Count Data | 50 |

271 Distributions for Interarrival Times | 52 |

272 Renewal Processes | 54 |

273 Gamma Count Distribution | 56 |

274 Duration Mixture Models | 59 |

Poisson Regression | 63 |

313 Ordinary Least Squares and Other Alternatives | 65 |

314 Interpretation of Parameters | 70 |

315 Period at Risk | 74 |

32 Maximum Likelihood Estimation | 77 |

323 NewtonRaphson Algorithm | 78 |

324 Properties of the Maximum Likelihood Estimator | 80 |

325 Estimation of the Variance Matrix | 82 |

326 Approximate Distribution of the Poisson Regression Coefficients | 83 |

327 Bias Reduction Techniques | 84 |

33 PseudoMaximum Likelihood | 87 |

331 Linear Exponential Families | 89 |

332 Biased Poisson Maximum Likelihood Inference | 90 |

333 Robust Poisson Regression | 91 |

334 NonParametric Variance Estimation | 95 |

335 Poisson Regression and LogLinear Models | 97 |

336 Generalized Method of Moments | 98 |

34 Sources of Misspeciﬁcation | 102 |

342 Unobserved Heterogeneity | 103 |

343 Measurement Error | 105 |

344 Dependent Process | 107 |

346 Simultaneity and Endogeneity | 108 |

347 Underreporting | 109 |

349 Variance Function | 110 |

35 Testing for Misspeciﬁcation | 112 |

352 Regression Based Tests | 118 |

354 Tests for NonNested Models | 120 |

36 Outlook | 125 |

Unobserved Heterogeneity | 127 |

412 Partial Effects with Unobserved Heterogeneity | 128 |

413 Unobserved Heterogeneity in the Poisson Model | 129 |

414 Parametric and SemiParametric Models | 130 |

421 Gamma Mixture | 131 |

423 LogNormal Mixture | 132 |

43 Negative Binomial Models | 134 |

431 Negbin II Model | 135 |

432 Negbin I Model | 136 |

434 NegbinX Model | 137 |

44 Semiparametric Mixture Models | 138 |

442 Finite Mixture Models | 139 |

Sample Selection and Endogeneity | 143 |

511 Truncated Count Data Models | 144 |

513 Censored Count Data Models | 146 |

514 Grouped Poisson Regression Model | 147 |

533 Endogeneity and Mean Function | 159 |

534 A TwoEquation Framework | 161 |

535 Instrumental Variable Estimation | 162 |

536 Estimation in Stages | 165 |

54 Switching Regression | 167 |

541 Full Information Maximum Likelihood Estimation | 168 |

542 MomentBased Estimation | 170 |

543 NonNormality | 171 |

Zeros in Count Data Models | 173 |

62 Zeros in the Poisson Model | 174 |

622 TwoCrossings Theorem | 175 |

623 Effects at the Extensive Margin | 176 |

624 MultiIndex Models | 177 |

63 Hurdle Count Data Models | 178 |

631 Hurdle Poisson Model | 181 |

632 Marginal Effects | 182 |

633 Hurdle Negative Binomial Model | 183 |

635 Unobserved Heterogeneity in Hurdle Models | 185 |

636 Finite Mixture Versus Hurdle Models | 186 |

637 Correlated Hurdle Models | 187 |

64 ZeroInﬂated Count Data Models | 188 |

642 ZeroInﬂated Poisson Model | 189 |

643 ZeroInflated Negative Binomial Model | 191 |

65 Compound Count Data Models | 192 |

651 MultiEpisode Models | 193 |

653 Count Amount Model | 196 |

654 Endogenous Underreporting | 197 |

66 Quantile Regression for Count Data | 199 |

Correlated Count Data | 203 |

711 Multivariate Poisson Distribution | 205 |

712 Multivariate Negative Binomial Model | 210 |

713 Multivariate PoissonGamma Mixture Model | 212 |

714 Multivariate PoissonLogNormal Model | 213 |

715 Latent PoissonNormal Model | 216 |

716 MomentBased Methods | 217 |

717 Copula Functions | 219 |

72 Panel Data Models | 220 |

721 Fixed Effects Poisson Model | 222 |

722 Momentbased Estimation of the Fixed Effects Model | 225 |

723 Fixed Effects Negative Binomial Model | 227 |

724 Random Effects Count Data Models | 228 |

725 Dynamic Panel Count Data Models | 230 |

73 TimeSeries Count Data Models | 232 |

Bayesian Analysis of Count Data | 241 |

81 Bayesian Analysis of the Poisson Model | 242 |

82 A Poisson Model with Underreporting | 245 |

83 Estimation of the Multivariate PoissonLogNormal Model by MCMC | 247 |

84 Estimation of a Random Coefficients Model by MCMC | 248 |

Applications | 251 |

92 Crime | 252 |

94 Health Economics | 254 |

95 Demography | 257 |

96 Marketing and Management | 260 |

97 Labor Mobility | 261 |

971 Economics Models of Labor Mobility | 262 |

972 Previous Literature | 263 |

973 Data and Descriptive Statistics | 265 |

974 Regression Results | 269 |

975 Model Performance | 272 |

976 Marginal Probability Eﬀects | 274 |

977 Structural Inferences | 278 |

Probability Generating Functions | 281 |

GaussHermite Quadrature | 285 |

Software | 288 |

Tables | 291 |

299 | |

321 | |

326 | |

### Common terms and phrases

ˆβ alternative approach assumption asymptotic bivariate censoring Chap coefficients conditional distribution conditional expectation constant correlation count data distribution count data models covariance matrix denote density function dependent variable derive discussed distribution function endogenous example exogenous explanatory variables exponential family fixed effects gamma distribution given hurdle model hurdle Poisson independent instance integer job changes joint distribution likelihood function linear exponential family linear model log-likelihood function marginal distribution marginal probability marginal probability effects maximum likelihood estimator mean function methods Mixture Models multivariate negative binomial distribution negative binomial model Negbin II model non-negative normal distribution number of events number of job observed obtained outcome overdispersion Poisson distribution Poisson model Poisson regression model Poisson-log-normal model probability function probability generating function quantile random variable regressors restrictions sample Santos Silva standard errors truncated underdispersion unobserved heterogeneity variance function vector Winkelmann zero

### Popular passages

Page 318 - Winkelmann, R. and KF Zimmermann (1995), Recent Developments in Count Data Modeling: Theory and Applications, Journal of Economic Surveys 9,1- 24.

Page 302 - Medicine 20, 3667-3676. van den Broek, J. (1995). A score test for zero inflation in a Poisson distribution.

### References to this book

Univariate Discrete Distributions Norman L. Johnson,Adrienne W. Kemp,Samuel Kotz Limited preview - 2005 |

Multivariate Statistical Modelling Based on Generalized Linear Models Ludwig Fahrmeir,Gerhard Tutz No preview available - 2001 |